对某些积分器而言，积分变量可以不是时间。
For some integrators, the variable of integration may be other than time.
以椭圆的离心角为积分变量，得到椭圆柔性铰链的转角计算的积分公式，推导出椭圆柔性铰链的刚度表达式。
In this paper, an integral formula for elliptical flexure hinge is deduced by introducing centrifugal Angle as the integral variable.
在一般教材中二重积分变量代换公式的证明通常采用几何的方法，也有部分数学分析教材给与了严格的分析证明，但证明不便直观的几何说明。
With proper method of variable substitution, the soluble method to two kinds of inegrable equation of Riccati form is found, and the general integral expressions are given.
然后就变成了一个单变量积分。
它们产生在一条曲线上，任何的积分都可以只用一个变量来表示，进行变量替换后，你得到一个，我们知道怎样计算的，单变量积分。
They take place in a curve.You express everything in terms of one variable and after substituting, you end up with a usual one variable integral that you know how to evaluate.
当计算一个单变量的积分时，你不会这样做的，不会分割出一个个小块，再对这些小块求和。
When you compute an integral in single variable calculus, you don't do that. You don't cut into little pieces and sum the pieces together.
这一个是线积分，可以用这里的办法来做，也就是用一个变量来表示出来。
This one is a line integral. So, you use the method to explain here, namely, you express x and y in terms of a single variable.
我要说的是取其一，积分得到包括另一变量的函数的结果，然后对结果求导，进行比较看得到什么。
But what I am saying is just take one of them, integrate, get an answer that involves function of the other variable, then differentiate that answer and compare and see what you get.
不要求掌握分部积分公式，但还是希望，有些人能在单变量积分中记住它。
Well, I don't expect that you would need integration by parts, although I still hope that some of you remember it from single variable calculus.
这可能是周四考试的考点，如果周四考试遇到此类问题，在做了变量替换之后，积分边界变得很简单的。
So, probably the one that will be there on Thursday, if there's a problem about that on Thursday, it will be a situation where the bounds that you get after changing variables are reasonably easy.
我强烈建议,如果你忘记了单变量积分的知识，现在就是一个很好的复习机会。
OK, so, yeah that's a strong suggestion that if you've forgotten everything about single variable calculus, now would be a good time to actually brush up on integrals.
并且这个积分常数,取决于剩下的变量,它可能是y的函数，或者在空间中是y和z的函数。
And that integration constant typically depends on the remaining variables that might be y or equal in space y and z.
因为这个变量，和积分的变量不一样。
And it's because this variable here is not the same as the variables on which we are integrating.
那么你要做一个以t为变量的积分。
只要它是这种情况,记住y只是积分中的变量。
As long as that's the case Remember y is just a variable of integration here.
不是变量，这是在一个圆上。，R，is，not，a，variable。，You，are，on，the，circle。,这个是二重积分，如果你们这么做的话，在圆盘上，如果你们用极坐标的话,就需要用到R和θ
R This one is a double integral. So, if you are doing it, say, on a disk, you would have both R and theta if you're using polar coordinates.
这边需要对两个变量做积分。
在平面上的线积分中，有两个变量，可以通过了解曲线的形成规律，从而去掉一个变量。
In the line integral in the plane, you had two variables that you reduced to one by figuring out what the curve was.
所有的这些都跟一个变量有关，也就可以来做单变量积分了，有问题么？
You express everything in terms of a single variable and then you do a usual single integral. Any questions about that?
另一件关于二重积分的是，我们已经讲过了，如何做更复杂的变量变换。
OK, now another thing we've seen with double integrals is how to do more complicated changes of variables.
其中很多内容无疑是非常前沿——整合语言学，还有整合积分学，这是一种取代变量的数学视角。
Part of it does seem definitely new — an integral semiotics, as well as an integral calculus, a form of mathematics that replaces variables with perspectives.
它是一个面积分，所以这里需要两个变量。
It is a surface integral, so we need to have two variables in there.
但是这仍然是一个线积分，代入计算就会发现，这仍然会变为单变量积分。
But it is still a line integral so it is still going to turn into a single integral when you plug in the correct values.
微积分是数学史上的一次重大突破，因为她使连续变量可以被处理。
Calculus was a mathematical breakthrough, because it dealt with continuously varying quantities.
在复变量函数理论中广泛使用线积分。
Line integrals are used extensively in the theory of functions of a complex variable.
指出了椭球形区域上三重积分的一科变量替换方法，并说明了其应用。
This paper points out the variable substitution method of triple integral of elliptical volume and examples its applications.
将响应面方法计算的位移对设计变量的敏度与莫尔积分方法的近似显式进行了对比。
The sensitivity of displacement with respect to design variables was compared to the approximate explicit form given by Mohr integration.
当随机变量的均值为零时，这个解在第一象限上的积分值可简化为一个已知的结论。
When the mean of random variables is zero, the solution is shown to reduce a known result for the value of the integral over the first quadrant.
本文给出了具有变量可分离核的积分方程解的存在条件和初等求解法。
This paper gives existence and uniqueness condition of solutions of the integral equation with variable separable kernel, and it's elementary solutions.
本文给出了具有变量可分离核的积分方程解的存在条件和初等求解法。
This paper gives existence and uniqueness condition of solutions of the integral equation with variable separable kernel, and it's elementary solutions.
应用推荐